Neuron Dynamics Notes

8/4/2019 Neuron Dynamics Notes

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Introduction to Cognitive Science for Mathematical Scientists

Cognitive Science 050.313/613

Notes on Neuron Dynamics: The Action Potential

Neural Currents I: Analysis Summary

September 5, 2003

Equation Quantity Prerequisite

a.

b.

Hodgkin-Huxley Eqn:

Neural currents I

Voltage-dependent currentgates i

Cable Equation

I=gV Ohms Law

g =g1 +g2 Parallel conductances

I = CdV/dt Capacitance

Ei Resting potentials (c)

g(V, t) Gates: H-H model (f)

c. Ei = (kT/ze) ln([Xi]out/[Xi]in) Nernst Equation

Resting potentials Ei

Nernst-Planck Equation (d)

d.Nernst-Planck Equation

Currents from drift,diffusion

Jdrift ,Jdiff equations

f. Gate conductance asfunction of gating particlestates

Hodgkin-Huxley gate model

g.

sim. for m(t), h(t)

Time-dependence of gates Hodgkin-Huxley gate model:reaction kinetics

h. Voltage-dependence ofgates

Energy barrier model

Boltzmann distribution

Key:

q charge (coulombs)

I current (charge/sec= amperes)

J current density (charge/sec-cm2)

E electric field (force/charge)

V voltage = potential energy/charge (joules/coulomb) E = dV/dt

R resistance (to current flow; ohms V= IR

g conductance = 1/R J = E

conductivity (inherent medium conductance) g1+2 in parallel =g1 +g2

C capacitance I= C dV/dt

K,Na,L

( , )[ ]m m i ii

VJ C g V t V E

t =

= +

2

22m

i

a VJ

R x

=

[ ] [ ]

kTV XJ z X

x q x

= +

3Na Na( ) ( ) ( )g t g m t h t=

4K K( ) ( )g t g n t=

( ) [1 ( )] ( )n ndn t n t n tdt

=

(1 ) /0 ( ) zeV kT n nV e =

/0 ( ) zeV kT n nV e=


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2Introduction to Cognitive Science for Mathematical Scientists


Neural Currents II: Fundamentals of Electric Currents

September 5, 2003

(1)

Force, energya. Kinetic Energy K mv2 [kg-m2/s2 joule] because:b. dK/dt = mv dv/dt = v ma = vF by Newtons Second Law of Motion c. dK/dt = F dx/dtd. K= Fxe. Potential energy (work) U Fx [actually, UFdx, = vector inner product]f. Total energy K+ U because:g. = 0 Conservation of Total Energyh. At temperature T(in K, degrees Kelvin; k Boltzmanns constant):

Scale of molecular energy = kT k = 1.38 1023joule/K

(2) Momentuma. F= ma = m dv/dt = d (mv)/dt = dp/dt [kg-m/s2 newton] whereb. pmv is momentumc. Vector quantity: p = mv i.e., (p1, p2, p3) = m(v1, v2, v3) = (mv1, mv2, mv3)d. dp/dt = F i.e., (dp1/dt, ) = (F1, )e. p = Ft impulsef. Conserved: dpuniverse/dt = 0 why?g. Fon 1 from 2 = Fon 2 from 1 Newtons First Law of Motion h. d(p1+p2)/dt = Fon 1 from 2 + Fon 2 from 1 = 0 from d

(3) Electric charge, current, force, fields, potentiala. Electric charge, q (for electron, e)b. Coulombs law: 1 2

2

kq qF

r= [k = 1 iff q in coulombs]

c. In general: F = qE [E in volts/m]d. Potential Energy difference U

= Fx = qEx

e. Potential Energy differenceper unit chargevoltage dropV U/q = Ex [Vin joules/coulomb volts]

f. Current charge/sec: Ior I [coulombs/s amperes]g. Conservation of charge: iIi = 0 Kirchoffs Law ({Ii} = the currents into a point)h. Current density, current/cm2:JorJ

(4) Resistance (R), conductance (g)a. V= IR I=gV (g = 1/R) [R in ohms, ] Ohms Law. Resistor. Why?b. Current densityJ; IJAc. Mysterious fact:J= E = conductivityd. V= xE = xJ/ = xI/AIR; Rx/Ae. Why the mysterious fact? model of electrical conductance

i. key: scattering of e off lattice

v

p


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ii. e collides with molecule on average after time (giving momentum to the lattice)iii. hard sphere collision: outgoing direction of e is randomiv. let v be the velocity of the e coming out of last collision, time t agov. during time t gains momentum Ft = eEt

vi. momentum at end of time t is p = mv + eEtvii. average momentum of all electrons is p = mv + eEt = 0 + eE

viii. this is m(effective [drift] velocity of e cloud) mvix. v = (e/m)E (cm/sec)x. suppose density of es is n per cm3

xi. J current density (charge flowing through 1 cm2 in 1 sec) = (charge in a volume v cm3)= nev = ne2/mE

xii. J = E ne2/mxiii. Generalizes to other charge carriers and non-hard-sphere-collisions ( randomization time)

(5) Resistors in the combinatorial strategya. Wires: R = 0, V= 0b. Resistors in series

i. Vadds, Iin common, so ii. R12 = R1 + R2

iii. Check: (4)d says Rx/A ; x12 = x1 + x2 ;A, unchangedc. Resistors in parallel

i. Iadds, Vin common, so ii. g12 =g1 +g2

iii. Check: (4)d says Rx/A ;A12=A1 +A2 ; x, unchanged(6) Capacitance

a. Charge can accumulate on/in materialsb. E.g.: parallel conducting plates; capacitorc. Q = C V due to relation between Q and E, Coulombs lawd. I= dQ/dt = CdV/dt Iinto capacitore. The basic RCcircuit:

i. As current flows through a medium with a given R, C, equivalent to a parallel circuitii. V/R = I= CdV/dt because here Iis out from capacitor

iii. dV/dt = (1/)V; RCiv. V= V0et/ = time constant of exponential decay, V0V(t=0)

V

R

C


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Neural Currents III: Qualitative Analysis

September 5, 2003

(1) Backgrounda. Ions most important in biological currents: K+, Na+, Cl

, Ca++

; inside cell, other anions A

Typical mammalian cell:

Ca++ ClNa+ Na+ClCa++

A K+ K+ A

b. Space-charge neutrality normally holds (total +q) = (total q)(2) Ion concentrations

a. Are constant, even during signal propagation, to within .01%.b. Primary determinants:

i. Membrane is impermeable to anions (organic acids and proteins) A concentrated insideii. Membrane is highly permeable to K+

iii. Membrane is moderately permeable to Cliv. Axon: At resting voltages (~ 70 mV), membrane is very slightly permeable to Na+;

as voltage increases (above ~ 50 mV), membrane becomes increasingly permeable to Na+,becoming extremely permeable (high relative to K+) at peak of action potential (variation in

permeability: gNa from .05gK to 500gK.

c. Na-K pumpi. 3Na+ out for every 2K+ in

ii. not a major factor in determining most resting concentrations, or short-term behavior, but iii. essential over long term for enabling unequal concentrations to be maintained after action potentials,

which involve Na+, K+ currents that would eventually alter resting concentrations(3) Consequences at rest

a. Start with A, K+ and Cl concentrationsi. Space-charge neutrality outside [Cl]out = [K+]out

ii. Space-charge neutrality inside (given A inside) [Cl]in < [K+]inb. Cant have [K+]in < [K+]out

i. For contradiction, assume so. Then have [Cl]in < [Cl]out Cl diffusion current inwardii. therefore equilibrium Cl drift current outward

iii. E directed inward iv. K+ drift current inward; but if [K+]in < [K+]out then K+ diffusion is inward too: no equilibrium

c. Therefore must have [K+]in > [K+]out and i. must have V Vin < Vout 0,

net charge on inside of membrane, net +charge on outside: no overall charge in/outside and

ii. must have [Cl]out > [Cl]in .d. Now add Na+:

i. Low membrane permeability to Na+ means low Na+ current, just to cancel pump Na+ currentii. Pump sends Na+ outward (and K+ inward, opposing its gradient)

iii. Diffusion Na+ current must be inward, so iv. must have [Na+]out > [Na+]in


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(4) Dendritea. Permeability increased by neuro-transmitter-sensitive gates at synapseb. Concentration gradients drive additional ionic current, reducing membrane charge imbalancec. Drives V less negative/more positive at synapse;d. Same for Vnearby, since dendrite is conductinge. V, I, decay quickly (exponentially, length constant 1 mm)f. Vs add up at beginning of axon, depolarizing it to a degree dependent on stimulation

(5) Consequences: axon slightly depolarized (75 mVV50 mV) [indirectly by dendrites]a. Permeability of membrane to Na+ increases slightlyb. Resting concentration gradient drives a little more Na+ in through membranec. Reduces net charge on inside surface of membrane slightlyd. Drives V slightly less negative/more positivee. Decreases slightly inward electromotive force on K+ in membranef. Reduces electromotive resistance to outward K+ diffusiong. Increases outward K+ flow slightly, counteracting increased inward Na+ flow:h.

Negative feedback

(6) Consequences when moderately depolarized (when V 50 mV)a. Resting concentration gradient drives Na+ in through elevated permeabilityb. Reduces net charge on inside surface of membranec. Drives V less negative/more positived. Increases permeability of membrane to Na+ still further:e. positive feedbackf. Eventually inflow of Na+ causes net inflow of total current (driven by Na+ concentrations)g. Na+ permeability rapidly increases: spikeh. Vrepolarizes because

i. Na+ channels spontaneously de-activate after being open awhileii. Voltage-sensitive K+ channels open (delayed relative to Na+ channels), K+ current cancels

(7) Propagationa. As in dendrites, Vchanges at one point on axon cause Vchanges at adjacent pointsb. large Vchange nearby :

i. down the axon: opens Na+ gates, re-initiates spikeii. up the axon: Na+ gates deactivated, K+ gates still open, so no spike

c. Myelini. plugs leaky membrane, reduces capacitance: speeds passive propagation ( = 1/RC)

ii.

gaps nodes of Ranvier have gates where spike re-initiates(8) Gates

a. Complex molecules with multiple statesb. Some states open a channel, significantly increasing conductivitygc. States described by variables n, m, h in Hodgkin-Huxley gate model transitions between states is

stochastic: probabilistic model to be discussed later in course.

d. Transition probabilities depend on voltage across membrane (unequal charge density in moleculeresponds to electric field)


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Neural Currents IV: Equilibrium Analysis

September 5, 2003

(1)

Passive current flow: Nernst-Planck Equationa. Given:

i. Y, an ion of valence z(charge q = ze)ii. a membrane permeable to Y, with different concentrations [Y]in , [Y]out

b. Two pressures driving flow of Yacross the membrane:i. Electric field causes drift flux (in molecules/sec-cm2, not charge/sec-cm2)

Jdrift = z [Yz] V = mobilityii. Concentration gradient causes diffusion flux; Ficks Law:

Jdiff = D [Yz] D diffusion coefficientD = (kT/e) Einstein (1905; Brownian motion)

iii. Net result: Nernst-Planck equation

(2) Equilibrium potential of an ion Yz , Vm: Nernst Equation no net Ycurrent across membranea. Vm Vin Vout , the voltage difference where J = 0 (Jdiff and Jdrift cancel equilibrium); the resting

potential of Yi:

Nernst Equation

(62 mV / z) log10 ([Y]out/[Y]in) at body temperature, 37

Derivation (Assuming constant E = Vm/x):

0 = Jdiff +JdriftJdiff =Jdrift

Dd[Y]/dx = z [Y] dV/dx = z [Y] E

d[Y]/dx = c [Y] czE/D = zEe/kT

[Y] = k e c x

[Y]out/[Y]in = e cx

ln([Y]out/[Y]in) = cx = zeEx/kT

(kT/ze) ln([Y]out/[Y]in) = Ex = Vm

b. Since conductance is defined by Idrift =gVand this is cancelled by Idiffat the resting potential Ei:Idiff igEi

This is (approximately) constant because the ion concentrations are (approximately) constant.

c. For two permeable ions Yn and Zp (because there is one common Vm for all ions): Donnan equilibrium

(3) Constant Ycurrent through membrane: Goldman-Hodgkin-Katz (GHK) modela. Ions flow through cross-membrane protein molecules with aqueous poresb. Assumptions (for simple pores not complex, V-sensitive channels)

[ ] [ ]

kTV YJ z Y

x e x

= +

out

in

[ ]ln

[ ]i

ii

YktE

ze Y=

11

out in

in out

[ ] [ ]

[ ] [ ]

pnY Z

Y Z

=


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i. Ion currents across the membrane obey the Nernst-Planck equationii. Ions do not interact with each other

iii. E is constant in the membranec. GHK current equation:

where Pi iikT/le, i water-membrane partition coefficient for Yi , eV/kT, and

Derivation: Ii = zieJi (Ii in coulombs/cm2;Ji in molecules/cm2; l membrane thickness)

Ii = ai[Yi] bid[Yi]/dx Nernst-Planck: aiizi2eV/x = izi2eV/l biizikT

(I> 0 by def. when Iflows out through the membrane)

bid[Yi]/dx = Iiai[Yi] = y yIiai[Yi] change variable to y

dy/dx = dIi/dx ai d[Yi]/dx

= 0 ai (1/bi) y

= ai (y/bi)

ai/bi = (zie/kT)V/l = zi/l

k = y(0) = Ia[Y]x=0 [Y]x=0 = i[Y]in [Y]x=l = i[Y]out (definition of i)

Iia[Y]x=l = y(l)

where aii = izi2ieV/l = Pizi2e

Note: Can view as the net result of Iout= [Y]in , Iin= ez[Y]out, Pz2e/(1 ez)d. Multiple ions, each Ii= constant, Itotal = 0 (equilibrium)

i. Itotal = iIi , where Ii is given in (c), by the assumption that currents are independentii. The resting potential of a cell with ions Yi {K+, Na+, Cl} and membrane permeabilities Pi

GHK voltage equation

Derivation: For K+, Na+, and Cl, z2 = (1)2 = 1. Want V= kT/e = (kT/e )ln(e)

0 = Itotal = IK + INa + ICl

2 2in out out in

[ ] [ ] [ ] [ ]

1 1

i i

i i

z zi i

i i i i iz z

Y Y e Y Y eI P z e P z e

e e

= =

-

/i ia x by ke= /iz x lke=

0( [ ] )

i iz zi i xke I a Y e== =

out in(1 ) [ ] [ ]i i

z zi i i i iI e a Y a Y e =

2

04

[ ] [ ]2

i iz zi i i i x l i i x

b b acI I e a Y a Y e

a= =

=

out in

[ ] [ ]

1

i

i

zi i

i i i z

Y Y eI a

e

=

2 2out in in out

[ ] [ ] [ ] [ ]

1 1

i i

i i

z zi i i i

i i i i iz z

Y Y e Y Y eI P z e P z e

e e

= =

K out Na out Cl inrest

K in Na in Cl out

[K ] [Na ] [Cl ]ln

[K ] [Na ] [Cl ]

P P PkTV

e P P P

+ +

+ + + +

=+ +

K Na Cl

+ + + + - - in out in out out in

K Na Cl

0

[K ] [K ] [Na ] [Na ] [Cl ] [Cl ]

1 1 1

I I I

e e eP P P

e e e

= + +

= + +

( ) ( ) ( )+ + + + - - K in out Na in out Cl out in0 [K ] [K ] [Na ] [Na ] [Cl ] [Cl ]P e P e P e = + +

( )+ + - + + - K in Na in Cl out K out Na out Cl in[K ] [Na ] [Cl ] [K ] [Na ] [Cl ]P P P P P P e+ + = + ++ + -

K out Na out Cl in+ + -

K in Na in Cl out

[K ] [Na ] [Cl ]/ ln( ) ln

[K ] [Na ] [Cl ]

P P PkTV kT e e

e P P P

+ += =

+ +


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Note 1: with a single ionic species, Itotal = Ii , this becomes the Nernst Equation Note 2: one of the ion species can be an ion pump effect of pump on Vrest (10%)

References:

Kandel, Eric R., Schwartz, James H., and Jessell, Thomas M. 1991. Principles of Neural Science. New York: Elsevier. Third

Edition, Chapter 6.

Johnston, Daniel, and Wu, Samuel Miao-Sin. 1995. Foundations of Cellular Neurophysiology. Cambridge, MA: MIT Press.


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Neural Currents V: Propagating Action Potentials

September 5, 2003

(4)

Parametersa. a = radius of cylinder;A = a2 = cross-sectional area of cylinderb. Ri = specific intracellular resistivity (-cm)

Rix resistance to axial flow of a length x of cable = Rix/A ( = -cm cm/cm2)

c. Rm = specific membrane resistivity (-cm2)Rm

x resistance to trans-membrane (radial) flow of a length x of cable = Rm/2ax ( = -cm2 /cmcm)

d. Cm = specific membrane capacitance (F/cm2)Cm

x capacitance of membrane of a length x of cable = Cm2ax (F = F/cm2 cm2)

e. Ji = interior (axial) current densityIi =Ji A interior (axial) current

f. Jm = trans-membrane (radial) current densityIm

x radial current along a length x =Jm 2ax

(5) Cable equation [See Fig 4.6, (4.4.11), p. 63, Johnston and Wu]a. Ohms Law along the cable axis:

b. Kirchoffs law for current split between axial current and trans-membrane current

c. Combining:

d. Independently, if E is the equilibrium (reversal) potential for the ion in question*:

e. Combining:

f. This is the cable equation, which can be rewritten

where the characteristic length and time are respectively:

= = =

= V

x V I R I R x AV

xJ Rm i i

x

i i i i /

0 2 2=

+ =

+

=

I

xx I

JA x J a x

J

xJ ai

m

i

m

i

m

( / )

=

= =

2

2

2

22

2

V

x

J

x R J R a J

a

R

V

x

ii m i m

i

( / )

drift diff

1( )

x x x x x m mm C m x x

m m

m m mm m m m m m

m m m

V V EI I I I C

t R R

V V VEJ C C g V E g

t R R t R

= + + = +

= + = +

2

2( )

2m

m m m m

i

Va VJ C g V E

R x t

= = +

22

2

=

+ V

x

V

tV Em

mm( )

aR

RR Cm

i

m m m2


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g. Solutions: A bear. [See Johnston & Wu, Sect. 4.4.2, pp. 6684]h. *Note concerning variation of diffusion current with changing membrane conductance

i. As gates open/close, mobility of ions through them goes up/down.ii. Diffusion current is proportional to D which is proportional to ; drift current is proportional to /g

which is proportional to .

iii. The diffusion current at resting potential is giEi, opposing the drift current; and asg changes with V,the diffusion current changes proportionally, remaining equal togiEi

(6) HodgkinHuxley equation [See Fig 3.4, (3.4.3) p. 48; Fig 6.8, p. 149, Johnston & Wu]a. Cable equation with multiple ion species Xi, each with its own conductancegi and equilibrium potential

Ei

b. In terms of conductancesgi = 1/Rm,i : (L leak current Cl current)

(7) Equations forg(V, t)? Probabilistic model of ionic channels.(8) How to solve? In general: iterate difference equation; numerical integration. See Fig. 6.13

J CV

tg V t V E m m i i

i

=

+ = ( , )[ ]K,Na,L

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